A product formula for period integrals
For \(n \geq 2\) let \(\lambda_ i\) be \(n\) distinct complex numbers and \(B^{(i)}\) be \(n\) \((r \times r)\)-matrices of complex numbers. Let \(P\) denote the differential operator \(\sum^ n_{i = 1} B^{(i)} / (x - \lambda_ i) dx_ i\). The main result of the paper is an exact formula for the determinant of the period mapping between the de Rham cohomology with compact support of \(\mathbb{C} - \{\lambda_ 1, \dots, \lambda_ n\}\) and the relative cohomology with local coefficients given by the differential equation \(dF = PF\) in terms of the local data of the \(\lambda_ i\)'s and \(\infty\) under some genericity conditions. This formula is an analogue of a formula of \textit{G. Laumon} [Publ. Math., Inst. Hautes Etud. Sci. 65, 131-210 (1987; Zbl 0641.14009)] in the context of \(\ell\)-adic cohomology. As a special case (\(n = 2\), \(r = 1\), \(\lambda_ 1 =0\), \(\lambda_ 2 = 1\), \(P = \left({\alpha \over t} + {\beta \over t - 1}\right)dt)\) one recovers Euler's formula \(B(\alpha \beta) = \Gamma(\alpha) \Gamma(\beta) /\Gamma(\alpha + \beta)\). The main idea of the proof is to show that the determinant of the Fourier transform is of the form \(c_ 0 t^ b \exp (kt)\) for some constants \(c_ 0\), \(b\), and \(k\).
scientific article; zbMATH DE number 562478
scientific article